${\mathrm{S}\mathrm{L}}_3$-laminations as bases for ${\mathrm{P}\mathrm{G}\mathrm{L}}_3$ cluster varieties for surfaces

In this paper we partially settle Fock-Goncharov’s duality conjecture for cluster varieties associated to their moduli spaces of $\mathrm{G}$-local systems on a punctured surface $\mathfrak{S}$ with boundary data, when $\mathrm{G}$ is a group of type $A_2$, namely ${\mathrm{S}\mathrm{L}}_3$ and ${\mathrm{P}\mathrm{G}\mathrm{L}}_3$. Based on Kuperberg’s ${\mathrm{S}\mathrm{L}}_3$-webs, we introduce the notion of ${\mathrm{S}\mathrm{L}}_3$-laminations on $\mathfrak{S}$ defined as certain ${\mathrm{S}\mathrm{L}}_3$-webs with integer weights. We introduce coordinate systems for ${\mathrm{S}\mathrm{L}}_3$-laminations, and show that ${\rm SL}_3$-laminations satisfying a congruence property are geometric realizations of the tropical integer points of the cluster $\mathcal{A}$-moduli space ${\mathcal{A}}_{{\mathrm{S}\mathrm{L}}_3,\mathfrak{S}}$. Per each such ${\mathrm{S}\mathrm{L}}_3$-lamination, we construct a regular function on the cluster $\mathcal{X}$-moduli space ${\mathcal{X}}_{{\mathrm{P}\mathrm{G}\mathrm{L}}_3,\mathfrak{S}}$. We show that these functions form a basis of the ring of all regular functions. For a proof, we develop ${\mathrm{S}\mathrm{L}}_3$ quantum and classical trace maps for any triangulated bordered surface with marked points, and state-sum formulas for them. We construct quantum versions of the basic regular functions on ${\mathcal{X}}_{{\mathrm{P}\mathrm{G}\mathrm{L}}_3,\mathfrak{S}}$. The bases constructed in this paper are built from non-elliptic webs, hence could be viewed as higher bangles' bases, and the corresponding bracelets’ versions can also be considered as direct analogs of Fock-Goncharov’s and Allegretti-Kim’s bases for the ${\mathrm{S}\mathrm{L}}_2$-${\mathrm{P}\mathrm{G}\mathrm{L}}_2$ case.